In mathematics, the Hecke algebra of a pair (G, K) of locally compact or reductive Lie groups is an algebra of measures under convolution. It can also be defined for a pair (g,K) of a maximal compact subgroup K of a Lie group with Lie algebra g, in which case the Hecke algebra is an algebra with an approximate identity, whose approximately unital modules are the same as K-finite representations of the pairs (g,K).
The Hecke algebra of a pair is a generalization of the classical Hecke algebra studied by Erich Hecke, which corresponds to the case (GL2(Q), GL2(Z)).